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Black Holes and Random Matrices

Jordan S. Cotler, Guy Gur-Ari, Masanori Hanada, Joseph Polchinski, Phil Saad, Stephen H. Shenker, Douglas Stanford, Alexandre Streicher, Masaki Tezuka

arXiv:1611.04650v3hep-thcond-mat.stat-mechquant-ph

TL;DR

The paper asks how the discrete spectrum and late-time fluctuations of large AdS black holes arise from a bulk perspective. Using SYK, analytically continued partition functions, correlation functions, numerics, and a double-scaling analysis, it finds late-time random-matrix behavior and formulates a conjecture for systems such as N = 4 SYM. The bulk interpretation remains an open challenge, and the extrapolation is subject to assumptions about intervening time scales and finite-N effects.

  • Problem

    The bulk origin of the discrete spectrum of black-hole microstates remains mysterious despite its clear boundary interpretation in AdS/CFT.

    Source: §1 · p. 3

  • Method

    The paper studies SYK using analytically continued partition functions, correlation functions, numerical calculations, and an exact double-scaling analysis, then compares the late-time behavior with random-matrix theory.

    Source: §1 · p. 6, §1.1 · p. 7

  • Results

    The spectral form factor and correlators exhibit late-time random-matrix behavior, while the early-time decay is determined exactly in a double-scaling limit.

    Source: §1.1 · p. 7

  • Takeaways & Limitations

    The results support a conjectural connection between late-time horizon fluctuations in large AdS black holes and random-matrix dynamics.

    Source: §10 · p. 42

    The SYK-to-holography extrapolation assumes no intervening nonperturbative time scales, while power-law-suppressed eigenvalue correlations can complicate the ramp at short times.

    Source: §6 · p. 26, §9 · p. 39

Abstract

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We argue that the late time behavior of horizon fluctuations in large anti-de Sitter (AdS) black holes is governed by the random matrix dynamics characteristic of quantum chaotic systems. Our main tool is the Sachdev-Ye-Kitaev (SYK) model, which we use as a simple model of a black hole. We use an analytically continued partition function $|Z(\beta +it)|^2$ as well as correlation functions as diagnostics. Using numerical techniques we establish random matrix behavior at late times. We determine the early time behavior exactly in a double scaling limit, giving us a plausible estimate for the crossover time to random matrix behavior. We use these ideas to formulate a conjecture about general large AdS black holes, like those dual to 4D super-Yang-Mills theory, giving a provisional estimate of the crossover time. We make some preliminary comments about challenges to understanding the late time dynamics from a bulk point of view.

1 Introduction

The paper uses late-time partition functions and correlation functions to probe the discrete spectrum of black-hole microstates, finding random-matrix behavior in SYK and extending the picture conjecturally to large AdS black holes.

1 Introduction

  • The motivation is that coarse-grained gravity predicts indefinite decay, whereas late-time deviations probe the discreteness of the black-hole spectrum and its nonperturbative scale.For holographic systems, the relevant scale is nonperturbative in 1/N.Source: §1 · p. 5
  • The paper argues that SYK's late-time behavior is closely related to random-matrix dynamics, consistent with quantum-chaotic level-spacing statistics at small energy differences.The relation is motivated by the conjecture that nearby energy levels in chaotic systems follow appropriate random-matrix ensembles.Source: §1 · p. 6

1.1 Summary of results

  • The authors conjecture analogous behavior for AdS5/CFT4 and N = 4 SYM, including an early-time decay followed by a long ramp-dominated period.Source: §1 · p. 6, §1.1 · p. 8
  • The spectral form factor develops a dip, linear ramp, and plateau matching Gaussian random-matrix behavior, with detailed shapes determined by the GUE, GOE, and GSE ensembles.The plateau characterizes spectral discreteness, while the ensemble choice controls the ramp's detailed shape.Source: §1.1 · p. 7
  • The paper develops numerical and analytic checks, including thermodynamic agreement with large-N calculations and an exact double-scaled density of states.Source: §1.1 · pp. 7–8
  • The early-time power-law decay is exact in a double-scaling limit through the Schwarzian sector, while the later ramp and plateau are argued to survive in the associated AdS2 black-hole limit.This provides a connection between early-time SYK dynamics and late-time random-matrix behavior.Source: §1.1 · p. 7
  • SYK correlators can also exhibit a ramp/plateau structure under specified conditions, and the paper checks this behavior numerically.Source: §1.1 · p. 7
  • For a single realization of the random couplings, the spectral form factor has large fluctuations even at large N, motivating procedures for comparison with non-ensemble-averaged theories.Source: §1.1 · p. 7

2 The Sachdev-Ye-Kitaev model

The SYK model is defined using Majorana fermions with random q-body couplings, focusing mainly on four-fermion interactions. Its Hamiltonian preserves fermion-number parity, producing even and odd charge-parity blocks.

2 The Sachdev-Ye-Kitaev model

  • SYK consists of N Majorana fermions in 0+1 dimensions with random q-fermion couplings, and this work mainly studies q = 4.Source: §2 · p. 9
  • The four-fermion coupling tensor is completely antisymmetric, with independent Gaussian entries of variance σ^2 = 3! N^−3 J^2.Source: §2 · p. 9
  • The Hamiltonian preserves charge parity rather than fermion number, so the Hilbert space splits into even- and odd-parity blocks.Source: §2 · p. 9

3 Spectral form factor

The SYK spectral form factor exhibits a slope, dip, linear ramp, and plateau, with late-time behavior closely matching random-matrix predictions. These structures probe spectral correlations at different energy separations and depend on the Hamiltonian’s symmetry class.

3 Spectral form factor

  • The spectral form factor is evaluated using annealed disorder averages, which require finitely many replicas and remain within a percent of quenched results in the reported numerical studies.The numerical calculations fully diagonalize independently sampled SYK Hamiltonians before averaging.Source: §3 · p. 10
  • SYK’s spectral form factor drops through a slope to a dip, rises approximately linearly in a ramp, and joins a late-time plateau set by the spectrum’s long-time average.For N = 34 and β = 5, the plateau occurs around tJ = 3 × 10^4; the plateau height scales as 2Z(2β)/Z(β)^2 ∼ e^-aS when the spectrum is twofold degenerate.Source: §3 · pp. 10–11, Figure 1 · p. 11
  • At late times SYK and GUE display the same dip–ramp–plateau structure, although their early-time decays differ because their eigenvalue densities and expansion parameters differ.Beyond the dip, the comparison is controlled by fine-scale eigenvalue correlations rather than the total spectral width.Source: §3 · p. 12, Figure 2 · p. 13
  • The ramp arises from long-range spectral rigidity: repulsion between widely separated eigenvalues produces logarithmic energy correlations, anticorrelation below the plateau, and approximately linear growth.The spectral form factor therefore probes pair correlations at larger separations than nearest-neighbor spacing statistics.Source: §3 · pp. 11–13, §3.1 · p. 14

3.1 The ramp and the eightfold way

  • Particle-hole symmetry organizes the relevant RMT ensemble by N mod 8: GUE applies for N mod 8 = 2 or 6, GOE for 0, and GSE for 4.The anti-linear symmetry acts differently on charge-parity sectors, determining the corresponding ensemble statistics.Source: §3.1 · p. 14
  • Nearest-neighbor statistics for SYK with N = 30 and 32 show excellent agreement with RMT, while the spectral form factor supplies complementary information about more widely separated levels.The RMT reference curves use unfolded spacings and exact diagonalization data with L = 12870.Source: §3 · p. 12, §3.1 · p. 14, Figure 3 · p. 15
  • The ramp shape follows the SYK symmetry class: GUE, GOE, and GSE produce distinct connections to the plateau, including a sharp corner, smooth join, or kink.For N = 34, the fitted ramp follows the expected GUE behavior g(t) ∼ t^1 to within a few percent; kinks at N = 20 and 28 signal GSE behavior.Source: §3.1 · p. 15
  • As N increases, the dip time grows rapidly while the plateau time grows faster, making the ramp increasingly prominent and suggesting that it survives in the large-N theory.The plateau heights also reflect degeneracy: they are 2/L when N mod 8 ≠ 0 and 1/L when N mod 8 = 0 at β = 0.Source: §3.1 · pp. 16–17, Figure 4 · p. 16

4 Thermodynamics of the SYK model

The section numerically extrapolates SYK thermodynamics to large N and finds excellent agreement with analytic large-N results for energy and entropy.

4 Thermodynamics of the SYK model

  • Large-N extrapolations of SYK thermodynamics agree excellently with existing analytic calculations, including the leading O(1/N) correction.The comparison serves as a cross-check on both the numerical results and the analytic calculations.Source: §4 · p. 17
  • Finite-N thermodynamic quantities are obtained by exact diagonalization and extrapolated at fixed temperature using a degree-two polynomial in 1/N.The leading coefficient gives the infinite-N result, while the next coefficient gives the 1/N correction.Source: §4 · p. 18
  • The extrapolated mean energy is almost indistinguishable from the exact large-N result obtained by numerically solving the Schwinger-Dyson equations.Figure 5 compares exact-diagonalization data, an eight-N three-term 1/N extrapolation, and the direct large-N solution.Source: Figure 5 · p. 18, §4 · p. 18
  • The low-temperature energy fit contains a constant term, a linear term, quadratic and cubic terms, and a fitted T^3.77 contribution.The reported fit is E/N = −0.04 −0.0025T + 0.22T^2 −0.52T^3 + 0.37T^3.77.Source: §4 · p. 19
  • The extrapolated coefficients give c within fifteen percent of its expected value and a fitted value of 1.6 versus the expected value 3.The coefficient a is suppressed at infinite N as expected, while the fitted 1/N value is fairly close to the expected value.Source: §4 · p. 19
  • The extrapolated entropy agrees excellently with a direct infinite-N calculation and correctly captures the large-N zero-temperature entropy.At fixed finite N, the entropy instead vanishes as T → 0.Source: §4 · pp. 19–20, Figure 6 · p. 20

5 Spectral form factor in random matrix theory

The section derives the random-matrix spectral form factor using the GUE, explaining its decay, ramp, plateau, and dip-time scales. It shows that the ramp is perturbative in RMT but becomes nonperturbative in SYK because the Hilbert-space dimension grows exponentially with N.

5 Spectral form factor in random matrix theory

  • The large-L saddle is the Wigner semicircle law, with physical eigenvalue density L times the unit-normalized density and average level spacing of order 1/L.The GUE calculation diagonalizes the matrix and replaces individual eigenvalues by a density whose normalization is constrained.Source: §5 · p. 21
  • Before the dip time, the spectral form factor is dominated by the disconnected part, whose late-time decay is proportional to 1/t^3.This power follows from the square-root vanishing of the mean eigenvalue density near the semicircle edge and holds at finite temperature.Source: §5 · p. 22

5.1 The ramp and the dip time

  • The connected spectral form factor behaves as gc(t) ∼ t/L^2 for 1 ≪ t ≤ L, making the ramp accessible through perturbation theory.Beyond the dip time, the full and connected spectral form factors are nearly equal and exhibit the ramp/plateau structure.Source: §5.1 · pp. 22, 24
  • The GUE spectral form factor has a linear ramp extending to plateau time 2L, followed by a constant plateau.The ramp lies below the plateau because eigenvalue anticorrelations produce the relevant negative contribution.Source: §5.1 · pp. 22–23
  • The derivation is controlled near the center of the semicircle and remains valid in 1/L perturbation theory only up to times t ∼ εL.The analysis focuses on eigenvalues near the center, where the mean density is L/π.Source: §5.1 · pp. 22–23
  • Equating the early-time slope with the ramp gives a dip time exponential in the entropy, while the plateau-to-dip ratio is also exponential, preserving the ramp at large L.The slope and ramp arise from disconnected and connected contributions, respectively.Source: §5 · p. 22, §5.1 · p. 24
  • In RMT the ramp is a perturbative effect of order 1/L^2, whereas the plateau requires nonperturbative sine-kernel or instanton physics.For GUE the ramp is the full perturbative result; other ensembles receive higher-order perturbative corrections.Source: §5.1 · p. 24
  • For SYK, L = 2^(N/2), so perturbative RMT effects scale as e^(−2aN), while nonperturbative RMT effects scale as exp(−e^(aN)).Thus effects perturbative in 1/L are already nonperturbative in the SYK 1/N expansion.Source: §5 · p. 21, §5.1 · p. 24

6 Spectral form factor in the SYK model

The SYK spectral form factor combines an analytically controlled early-time decay with a late-time ramp and plateau described using random-matrix spectral correlations. Numerical comparisons support this picture, while finite-N corrections and unresolved spectral-edge effects limit confidence in the crossover estimate.

6 Spectral form factor in the SYK model

  • A spectral-rigidity ansatz combines local GUE statistics across narrow energy bands, producing a smoothed ramp that joins the plateau at tp = e^S(2β).The derivative approaches zero at the join, giving a smooth C1 transition despite kinks in individual-band ramps.Source: §6 · pp. 25–26
  • Applying the ramp formula to SYK is conditional because 1/N^q correlations, including sample-to-sample Hamiltonian-scale fluctuations, may dominate the ramp contribution at short times.The authors therefore treat the resulting spectral-form-factor curve as a reasonable but not fully controlled picture.Source: §6 · pp. 26, 28, 29
  • The Schwarzian sector is one-loop exact in the triple-scaled limit, fixing the low-energy density of states and the early-time power-law decay.The double- and triple-scaled limits isolate the Schwarzian contribution and yield an exact density-of-states result for that sector.Source: §6 · pp. 26–27
  • Equating the early decay and ramp estimates gives a dip time td ∼ e^(Ns0/2), while the plateau time is exponentially later.The analysis also establishes td(β) ≤ td(β = ∞) ≤ tp(β = ∞) = e^(Ns0).Source: §6 · pp. 28–30
  • Numerical evaluation at N = 34 agrees reasonably with exact diagonalization, but the ramp and plateau differ by order-one factors because the large-N free energy omits a proper one-loop correction.The discrepancy is amplified in the ramp and plateau through repeated division by the incorrect partition-function factor.Source: §6 · p. 28, Figure 7 · p. 29

7 Correlation functions

SYK two-point functions exhibit ramp-and-plateau behavior controlled by the model’s N mod 8 symmetry pattern. ETH and random-matrix reasoning explain both the selection rules and the expected ramp and plateau scales, with numerical figures supporting the predictions.

7 Correlation functions

  • The plateau vanishes when N/2 is even because charge-parity sectors are non-degenerate, whereas a particle-hole selection rule permits it only when N mod 8 = 2.The plateau condition is expressed equivalently as η(N) = 1.Source: §7 · pp. 30–31
  • When N/2 is odd, degenerate sectors effectively form one random-matrix sector, so factorized eigenvalue and eigenstate averages produce the ramp through the spectral form factor.The matrix-element factor is smooth at small energy separation, while the spectral-density correlator supplies the ramp.Source: §7 · p. 31
  • For N mod 8 = 2, the two-point function has both a ramp and a non-zero plateau; for N mod 8 = 6, it has a ramp but no plateau; for N mod 8 = 0, 4, it has neither.Figures 8 and 9 numerically support these symmetry-dependent combinations.Source: §7 · p. 32, Figure 9 · p. 33
  • ETH estimates the correlator plateau height as order 1/L, matching the spectral-form-factor plateau scale.For N = 18, the correlator plateau is approximately 0.0075 versus approximately 0.004 for the spectral form factor.Source: §7 · p. 32

7.1 The ramp in more general theories

  • Under ETH and an assumed RMT ramp, general chaotic theories have a plateau from diagonal matrix elements and a ramp from off-diagonal elements of order 1/L.After suitable time averaging, the correlator deviation from its plateau scales schematically as ⟨O(t)O(0)⟩ − Gp ∼ t/L^2.Source: §7.1 · pp. 33–34

8 Single realization of random couplings

A single SYK realization self-averages at early times but develops large late-time fluctuations, so the dip, ramp, and plateau are revealed by controlled sliding-time averaging. The required averaging window can remain parametrically shorter than the exponentially long ramp.

8 Single realization of random couplings

  • Before the dip, a typical SYK sample agrees well with the disorder-averaged spectral form factor and is therefore self-averaging.After the dip, typical realizations show large fluctuations and no longer approach the disorder-averaged value.Source: §8 · p. 35, Figure 10 · p. 35
  • The underlying dip, ramp, and plateau remain visible despite late-time fluctuations and can be exposed by averaging over a sliding time window.The window must be shorter than the ramp length to avoid smearing the features of interest.Source: §8 · pp. 35–36
  • To reduce fluctuations below 1/N^2, the required averaging window is no greater than N^4 even when tdecay is as large as 1, far shorter than the exponentially long ramp.Numerics show a systematic decrease of error after time averaging, though they are not conclusive.Source: §8 · pp. 36–37
  • On the plateau, autocorrelations decay on a typical time scale tdecay ∼ 1/N at large N and large β, followed after a few β by an integrable 1/(dt)^3 tail.The late power law is already exponentially suppressed and does not change the time-averaging estimate.Source: §8 · pp. 36–37
  • On the ramp, assuming RMT multipoint spectral correlations, autocorrelations decay as 1/(dt)^4 for most of the ramp and 1/(dt)^3 initially, implying tdecay ∼ 1.These integrable power laws allow averaging errors to be reduced while retaining exponentially many ramp data points.Source: §8 · p. 37

9 Conjecture about super-Yang-Mills

The paper conjectures that fine-grained SYM spectral statistics are random-matrix-like and estimates an exponentially long ramp between gravity-controlled decay and the plateau. The estimate depends on saddle-point assumptions and remains preliminary because several gravitational and spectral effects are unresolved.

9 Conjecture about super-Yang-Mills

  • The conjecture assumes SYM fine-grained eigenvalue statistics follow RMT and that no additional nonperturbative time scales intervene between gravitational times and the RMT regime.A nonzero θ term is used in the simplified discussion to break T symmetry and select GUE statistics.Source: §9 · p. 37, §9.1 · p. 40
  • Early-time saddle analysis gives |Z|^2 ∼ e^(cN^2/β^3) at t = 0, a much smaller thermal-AdS contribution after t ∼ β, and renewed growth to |Z|^2 ∼ e^(cN^2) at the AdS scale.These competing saddles determine the proposed early-time structure before the ramp.Source: §9 · pp. 38–39
  • The conjecture is not conclusive because saddle-point corrections can become large at t ∼ N^2, other saddles may matter, and the relevant saddle contours are unresolved.The discussion explicitly notes that a more careful analysis is required for some candidate saddles.Source: §9 · p. 39, §9.1 · p. 40

9.1 The ramp in SYM

  • The dip is tentatively associated with small but non-microscopic black holes of radius rs ∼ β, giving an exponential time with a parametrically small high-temperature coefficient.The dip estimate is sensitive to the assumed long-time slope; alternative slope behavior changes the coefficient while preserving a large dip-to-plateau hierarchy.Source: §9.1 · p. 41
  • At high temperature, the SYM plateau time is tp = e^(4c0N^2/(2β)^3), producing a parametrically long ramp after the dip.The ramp joins the plateau when the dominant energy satisfies S(E) = log t, with the plateau reached at tp = e^S(2β).Source: §9.1 · pp. 40–42
  • Independent global-symmetry sectors are expected to suppress ramp and plateau heights by powers of the entropy, while leaving the leading exponential time scales unchanged.The authors report confirmation of this sector decomposition in Dirac SYK with a U(1) charge.Source: §9.1 · p. 42

10 Discussion

The discussion argues that SYK and large AdS black holes exhibit late-time random-matrix behavior, while identifying unresolved bulk explanations and limitations of current saddle analyses.

10 Discussion

  • Numerics establish late-time random-matrix behavior, while the double-scaling analysis fixes early-time behavior and yields an exponentially late dip time and exponentially long ramp.The dip is estimated by intersecting the early- and late-time curves; the ramp extends to the asymptotic plateau.Source: §10 · p. 42
  • Early-time spin-glass or 1/N^q effects could steepen the decay and make the dip time earlier than the double-scaling estimate.These phenomena are absent in the double-scaling limit.Source: §10 · p. 42
  • The authors conjecture analogous behavior for general large AdS black holes, including those dual to 4D SYM, with a provisional ramp-time estimate from analytically continued AdS-Schwarzschild geometry.They again find an exponential hierarchy of scales.Source: §10 · p. 43
  • The authors suggest studying ramp precursors and the 1/N expansion to determine whether terms grow or instead exhibit factorial growth with exponentially small errors.Such signals could guide searches for ramp and plateau effects in SYM.Source: §10 · p. 43
  • The bilocal G, Σ functional integral is a rough bulk proxy that must contain ramp and plateau behavior, but their mechanism remains unresolved.The authors describe this as a continuing research problem.Source: §10 · p. 44
  • For q = 2, the ramp is a perturbative 1/N^2 effect and the plateau is an e^-N effect, whereas for interacting q > 2 the ramp is a 1/L^2, hence e^-N, effect.Auxiliary quantities can be computed by saddle point, but a single-saddle explanation of the interacting ramp faces obstacles.Source: §10 · p. 44
  • The N mod 8 eightfold pattern and its possible topological explanation remain open questions in the G, Σ formulation.The authors compare the hoped-for explanation to the topological term in the Haldane spin chain.Source: §10 · pp. 44–45
  • The plateau origin is mysterious, with a proposed but challenging mapping to effective random-matrix degrees of freedom at coupling 1/L.This would reinterpret the exp(−L) plateau as a standard nonperturbative effect.Source: §10 · p. 45

Note added

The note added corrects plotting and sampling errors, while the particle-hole appendix derives SYK spectral degeneracies from charge-parity sectors and N mod 8.

Note added

  • The corrected version reports that earlier plotting errors caused only a small approximately 1% uniform vertical shift and did not affect qualitative or quantitative conclusions.The errors concerned normalization in several figures and were corrected in this version.Source: Note added · p. 45
  • The revision also corrects sample construction, time-step choices, and reported sample counts for Figures 2, 4, 10, and 12.The corrected counts include 1,200,000 samples for N = 16 and 914 for N = 30 in the relevant figures.Source: Note added · pp. 45–46

A Particle-hole symmetry of SYK

  • Particle-hole symmetry maps fermion number Q to N_d − Q and acts within the two charge-parity sectors of the Hamiltonian.Its anti-linear structure determines whether the symmetry squares to +1 or −1.Source: A Particle-hole symmetry of SYK · pp. 46–47
  • When N_d is odd, particle-hole symmetry exchanges the even and odd charge-parity sectors, making them degenerate.This is the first case in the appendix’s sector-by-sector classification.Source: A Particle-hole symmetry of SYK · p. 47
  • When N_d is even and N_d mod 4 = 2, P^2 = −1 produces double degeneracy within each charge-parity sector.The anti-linear symmetry cannot map an energy eigenstate to itself in this case.Source: A Particle-hole symmetry of SYK · p. 47
  • When N_d mod 4 = 0, P^2 = 1 and no protected degeneracy follows.The resulting pattern is twofold degeneracy for N mod 8 ≠ 0 and no protected degeneracy for N mod 8 = 0.Source: A Particle-hole symmetry of SYK · p. 47

B The double-scaled SYK theory

The double-scaled SYK analysis derives spectral data from moments and Wick contractions, identifies a Q-Hermite distribution, and provides an exact triple-scaled result alongside a toy G, Σ reformulation.

B The double-scaled SYK theory

  • The double-scaled theory is solved by computing moments, organizing Wick contractions by crossings, and using the resulting moment distribution to obtain the spectrum.The relevant limit keeps q^2/N fixed, so shared fermions remain order one while higher correlations are suppressed by 1/N.Source: B The double-scaled SYK theory · pp. 48–49
  • The resulting distribution is related to the Q-Hermite measure with Q = e^-λ and is normalized to reproduce the total number of states.Its small-λ limit reproduces the large-q thermodynamics of [14].Source: B The double-scaled SYK theory · pp. 49–50
  • In the triple-scaled limit, the Schwarzian-sector result is exact rather than merely a one-loop approximation.The normalization agrees with the one-loop calculation of [14].Source: B The double-scaled SYK theory · pp. 50–51
  • An alternative G, Σ derivation integrates out σ exactly, yielding a Liouville-type action whose perturbation theory reproduces the moment computation through Gaussian integrals.This provides a second formulation of the double-scaled calculation.Source: B The double-scaled SYK theory · p. 51

C A toy G, Σ integral

  • The toy G, Σ integral rewrites the SYK calculation using bosonic bilocal variables and reproduces the fermionic answer after contour and integral manipulations.Numerical integration of rotated contours also gives the correct answer for several N and q values.Source: C A toy G, Σ integral · pp. 52–53
  • The toy integral has q saddle points, while insertions of degree greater than N vanish, reflecting the finite Grassmann structure encoded by the bosonic representation.The vanishing follows because at most N derivatives act before setting g = 0.Source: C A toy G, Σ integral · p. 53

D Subleading saddle points in the G, Σ variables

The appendix studies subleading G, Σ saddles: they become nearly degenerate in q = 2 but remain separated by a large-N action gap in q = 4.

D Subleading saddle points in the G, Σ variables

  • In q = 2, choosing the subdominant G− solution for selected Matsubara frequencies generates subleading saddles, which become almost degenerate with the dominant saddle at large βJ.Despite the near-degeneracy, an order-one imaginary-part difference prevents a soft mode; at βJ ≫ N, many saddles must be summed.Source: D Subleading saddle points in the G, Σ… · p. 54
  • For q = 4, numerical continuation from q = 2 finds corresponding subleading saddles, but their actions do not become degenerate at large βJ.Figure 11 compares standard and frequency-flipped solutions for q = 2 and q = 4 at βJ = 20π.Source: D Subleading saddle points in the G, Σ… · p. 54, Figure 11 · p. 55
  • The q = 4 subleading saddles retain a large N^2 action gap even at low temperature, explaining why they do not alter large-N thermodynamics.Preliminary evidence suggests the gap also persists for complex β.Source: D Subleading saddle points in the G, Σ… · p. 55
  • Additional q = 4 saddles that break time-translation invariance were found numerically, but their actions were larger than the standard saddle and they were not studied systematically.Their role therefore remains unsettled.Source: D Subleading saddle points in the G, Σ… · p. 55

E Saddle points and the q = 2 model

The q = 2 SYK model is nonchaotic but exhibits mini-ramp and mini-plateau behavior, while its correlation functions reveal how spectral correlations generate these features. Saddle-point analysis connects semicircle statistics to the ramp and replica-symmetry-breaking contributions to the plateau.

E Saddle points and the q = 2 model

  • The q = 2 model is equivalent to free fermions with a random mass matrix and has plateau time t_p ∼ N rather than t_p ∼ L.Its explicit random matrix produces mini-ramp and mini-plateau behavior despite the model not being chaotic.Source: E Saddle points and the q = 2 model · p. 56
  • Single-particle eigenvalues repel, but weak repulsion between multiparticle sectors sets the plateau scale through level spacing ∼1/N.This explains why the plateau time is expected to scale as t_p ∼ N.Source: E Saddle points and the q = 2 model · p. 56
  • The averaged unsquared correlation function has no ramp, whereas its connected quenched squared correlation function has both a ramp and a plateau.At β = 0, the relevant connected quantity equals g_c(t) for the skew-Hermitian ensemble.Source: E Saddle points and the q = 2 model · p. 57
  • The factorized partition function over frequencies simplifies derivatives of the sourced theory, enabling saddle-point evaluation of the quenched squared correlation function.Replica-symmetric saddles provide the dominant contribution, while fluctuations and replica-symmetry-breaking saddles generate the ramp and plateau, respectively.Source: E Saddle points and the q = 2 model · pp. 58–59
  • The skew-Hermitian calculation differs from the GUE result at order 1/N because of the Majorana reality constraint.The associated saddle-point treatment also leaves the direct analytical calculation of Z(t) and g(t) for future work.Source: E Saddle points and the q = 2 model · pp. 58–59

F On N −q vs. 2−N

The analysis explains how SYK diagrams can transition from perturbative N^-q suppression to the 2^-N scale governing the late-time ramp, while identifying obstacles to a simple saddle-point explanation. Numerical data support random-matrix ramp, plateau, and symmetry-pattern behavior, but finite N limits definitive scaling conclusions.

F On N −q vs. 2−N

  • Large-k graph contractions control short-distance eigenvalue correlations and the late-time linear ramp, whose GUE coefficient is 2^-N.In SYK, the leading graph class is broader and requires a 1/N rather than 1/L expansion, so the full sum is not analyzed systematically.Source: F On N −q vs. 2−N · pp. 59–60
  • For large k, SYK coefficients approach 2^(1−N), matching the GUE-like late-time ramp scale and suggesting universal local random-matrix statistics.Small-k terms remain suppressed by powers of N^-q, so the crossover occurs within the coefficient sequence rather than at a single perturbative order.Source: F On N −q vs. 2−N · p. 60

G Constraints on saddle point origins of the ramp

  • The ramp is exponentially small but more tractable than the plateau, which random-matrix analysis suggests may involve effects as small as exp(−e^N).The linear ramp is perturbative in random-matrix theory even though it may remain nonperturbative in SYK.Source: G Constraints on saddle point origins… · p. 61
  • A simple saddle-point explanation must accommodate complex-action saddles for N mod 8 and unsuppressed ramp and plateau fluctuations.Numerics find fluctuations of the same order as the mean, and no obvious mechanism ensures consistent backreaction between ⟨G⟩ and ⟨G^2⟩.Source: G Constraints on saddle point origins… · p. 61

H Data

  • The observed plateau-time scaling is faster exponential growth than the dip time, so the ramp length grows exponentially with N, although available N values cannot distinguish all scenarios.At N = 34, the fitted ramp power is consistent within a couple percent with GUE behavior g(t) ∼ t^1.Source: H Data · p. 62

H.1 Plots of g(t), gc(t), and gd(t)

  • The numerical curves show random-matrix-dependent plateau features: GSE peaks, GUE kinks, GOE smoothing, and mod-8-dependent plateau heights.The quantity g_d(t) becomes much smaller near the dip because edge fluctuations cancel, while finite sampling causes an apparent rebound afterward.Source: §H.1 · pp. 62–63, Figure 12 · p. 63

H.2 Dip time td, plateau time tp and plateau height

The numerical analysis estimates exponentially separated dip and plateau times and finds a ramp whose length grows exponentially with N. Plateau heights follow partition-function predictions with mod-8 degeneracy effects, while density-of-states plots show finite-N oscillations and non-Gaussian tails.

H.2 Dip time td, plateau time tp and plateau height

  • Dip-time fits are consistent with exponential growth, but power-law alternatives cannot be ruled out through N = 34 because the error bars are large.The dip-time uncertainty is estimated from the range where g(t) remains below 1.04 g_min.Source: §H.2 · pp. 63–64, Figure 13 · p. 64
  • At β = 5, the entropy-growth coefficient is numerically 0.249 ± 0.014, close to the theoretical coefficient 0.27 up to O(β^-2) corrections.This comparison supports the expected exponential plateau-time scaling.Source: §H.2 · p. 64
  • The ramp length grows exponentially in N because t_p ∼ e^(κ_p N) exceeds t_d ∼ e^(κ_d N), with κ_d < κ_p = S(2β).The authors caution that the available N values are insufficient for definitive statements.Source: §H.2 · p. 64
  • The plateau height agrees with Z(2β)/Z(β)^2, doubling for N mod 8 = 2, 4, or 6 because of eigenvalue degeneracy and remaining undoubled for N mod 8 = 0.At β = 0, the nondegenerate prediction equals 2^-N/2.Source: §H.2 · p. 65, Figure 14 · p. 65

H.3 Comparison of factorized and unfactorized quantities

  • Factorized and unfactorized quantities agree at early times when self-averaging applies, and g agrees at large N for all times, whereas g_c differs from its quenched version early.The comparison distinguishes annealed from quenched disorder averaging.Source: §H.3 · p. 65

H.4 Density of states ρ(E)

  • The normalized density of states shows almost periodic level-repulsion oscillations at small N, while large-N spectra approach a Gaussian in an L2 sense but retain exponentially many states in non-Gaussian tails.The energy scale of the tails is E ∼ N, beyond the Gaussian bulk width.Source: §H.4 · pp. 65–66, Figure 15 · p. 66
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